Chiral Perturbation Theory for η → 3π and η → π 0 γγ

(1)

ChPT for η→ 3π and

η→ π⁰γγ Johan Bijnens

Older reviews Model independent ChPT η→ 3π in ChPT η→ π⁰γγ Conclusions

Chiral Perturbation Theory for η → 3π and η → π

⁰

γγ

Johan Bijnens

Lund University

bijnens@thep.lu.se http://www.thep.lu.se/∼bijnens http://www.thep.lu.se/∼bijnens/chpt.html

Hadronic Probes of Fundamental Symmetries, Amherst, 6-8 March 2014

(2)

ChPT aspects of η → 3π and η → π

⁰

γγ

1 Older reviews

2 η → 3π: Some model independent comments/results Definitions

Experiment Why?

3 ChPT

4 η → 3π in ChPT LO

LO and NLO NNLO

5 η → π⁰γγ p⁴

Experiment Loops at p⁶, p⁸ All else

Distributions

6 Conclusions

(3)

Eta Physics Handbook: ETA01

(4)

ETA05: Acta Phys.Slov. 56(2006) No 3

Acta Physica Slovaca 56(2006) C. Hanhart

Hadronic production of eta-mesons: Recent results and open questions , 193 (2006) C. Wilkin, U. Tengblad, G. Faldt

The p d -> p d eta reaction near threshold , 205 (2006)

J. Smyrski, H.-H. Adam, A. Budzanowski, E. Czerwinski, R. Czyzykiewicz, D. Gil, D.

Grzonka, A. Heczko, M. Janusz, L. Jarczyk, B. Kamys, A. Khoukaz, K. Kilian, P. Klaja, J.

Majewski, P. Moskal, W. Oelert, C. Piskor-Ignatowicz, J. Przerwa, J. Ritman, T. Rozek, R. Santo, T. Sefzick, M. Siemaszko, A. Taschner, P. Winter, M. Wolke, P. Wustner, Z.

Zhang, W. Zipper

Study of the 3He-eta system in d-p collisions at COSY-11 , 213 (2006) M. Doring, E. Oset, D. Strottman

Chiral dynamics in gamma p -> pi0 eta p and related reactions , 221 (2006) H. Machner, M. Abdel-Bary, A. Budzanowski, A. Chatterjee, J. Ernst, P. Hawranek, R.

Jahn, V. Jha, K. Kilian, S. Kliczewski, Da. Kirillov, Di. Kirillov, D. Kolev, M. Kravcikova, T. Kutsarova, M. Lesiak, J. Lieb, H. Machner, A. Magiera, R. Maier, G. Martinska, S.

Nedev, N. Pisku\-nov, D. Prasuhn, D. Protic, P. von Rossen, B. J. Roy, I. Sitnik, R.

Siudak, R. Tsenov, M. Ulicny, J. Urban, G. Vankova, C. Wilkin The eta meson physics program at GEM , 227 (2006) K. Nakayama, H. Haberzettl

Photo- and Hadro-production ofÂ eta' meson , 237 (2006) S.D. Bass

Gluonic effects in eta and eta' nucleon and nucleus interactions , 245 (2006) P. Klaja, P. Moskal, H.-H. Adam, A. Budzanowski, E. Czerwinski, R. Czyzykiewicz, D.

Gil, D. Grzonka, M. Janusz, L. Jarczyk, B. Kamys, A. Khoukaz, K. Kilian, J. Majewski, W.

Migdal, W. Oelert, C. Piskor-Ignatowicz, J. Przerwa, J. Ritman, T. Rozek, R. Santo, T.

Sefzick, M. Siemaszko, J. Smyrski, A. Taschner, P. Winter, M. Wolke, P. Wustner, Z.

Zhang, W. Zipper

Correlation femtoscopy for studying eta meson production mechanism , 251 (2006) M.T. Pena, H. Garcilazo

Study of the np -> eta d reaction within a three-body model , 261 (2006) A. Gillitzer

Search for nuclear eta states at COSY and GSI , 269 (2006) A. Wronska, V. Hejny, C. Wilkin

Near threshold eta meson production in the dd -> 4He eta reaction , 279 (2006) M. Bashkanov, T. Skorodko, C. Bargholtz, D. Bogoslawsky, H. Calen, F. Cappellaro, H.

Clem\-ent, L. Demiroers, E. Doroshkevich, C. Ekstrom, K. Fransson, L. Geren,Â J.

Greiff, L. Gustafsson, B. Hoistad, G. Ivanov,Â M. Jacewicz, E. Jiganov, T. Johansson, M.M. Kaskulov,Â S. Keleta, O. Khakimova, I. Koch, F. Kren, S. Kullander,Â A. Kupsc, A. Kuznetsov, K. Lindberg, P. Marciniewski,Â R. Meier, B. Morosov, W. Oelert, C.

Pauly, Y. Petukhov,Â A. Povtorejko, R.J.M.Y. Ruber, W. Scobel, R. Shafigullin, B.

Shwartz, V. Sopov, J. Stepaniak, P.-E. Tegner,Â V. Tchernyshev, P.

Thorngren-Engblom, V. Tikhomirov, A. Turowiecki, G.J. Wagner, M. Wolke, A.

Yamamoto,Â J. Zabierowski, I. Zartova,Â J. Zlomanczuk

On the pi pi production in free and in-medium NN-collisions: sigma-channel low-mass

enhancement and pi0 pi0 / pi+ pi- asymmetry , 285 (2006) K. Schonning for the CELSIUS/WASA collaboration Production of omega in pd -> 3He omega at kinematic threshold , 299 (2006) J. Bijnens

Decays of eta and eta' and what can we learn from them? , 305 (2006) B. Borasoy, R. Nissler

Decays of eta andÂ eta' within a chiral unitary approach , 319 (2006) E. Oset, J. R. Pelaez, L. Roca

Discussion of the eta -> pi0 gamma gamma decay within a chiral unitary approach , 327 (2006)

C. Bloise on behalf of the KLOE collaboration Perspectives on Hadron Physics at KLOE with 2.5 fb^-1 , 335 (2006) T.Capussela for the KLOE collaboration

Dalitz plot analysis of eta into 3pi final state , 341 (2006) A. Starostin

The eta and eta' physics with crystal ball , 345 (2006) S. Schadmand for the WASA at COSY collaboration WASA at COSY , 351 (2006)

M. Lang for the A2- and GDH-collaborations

Double-polarization observables, eta-meson and two-pion photoproduction , 357 (2006) M. Jacewicz, A. Kupsc for CELSIUS/WASA collaboration

Analysis of eta decay into pi+ pi- e+ e- in the pd -> 3He eta reaction , 367 (2006) F. Kleefeld

Coulomb scattering and the eta-eta' mixing angle , 373 (2006) C. Pauly, L. Demirors, W. Scobel for the CELSIUS-WASA collaboration Production of 3pi0 in pp reactions above the eta threshold and the slope parameter alpha , 381 (2006)

R. Czyzykiewicz, P. Moskal, H.-H. Adam, A. Budzanowski, E. Czerwinski, D. Gil, D.

Grzonka, M. Janusz, L. Jarczyk, B. Kamys, A. Khoukaz, K. Kilian, P. Klaja, B. Lorentz, J.

Majewski, W. Oelert, C. Piskor-Ignatowicz, J. Przerwa, J. Ritman, H. Rohdjess, T. Rozek, R. Santo, T. Sefzick, M. Siemaszko, J. Smyrski, A. Taschner, K. Ulbrich, P. Winter, M.

Wolke, P. Wustner, Z. Zhang, Z. Zipper

The analysing power for the pp -> pp eta reaction at Q=10 MeV , 387 (2006) A. Nikolaev for the A2 and Crystal Ball at MAMI collaborations Status of the eta mass measurement with the Crystal Ball at MAMI , 397 (2006) B. Di Micco for the CLOE collaboration

The eta -> pi0 gamma gamma, eta/eta' mixing angle and status of eta mass measurement at KLOE , 403 (2006)

(5)

Older reviews Model independent

Definitions Experiment Why?

ChPT η→ 3π in ChPT η→ π⁰γγ Conclusions

Definitions: η → 3π

Reviews: JB, Gasser, Phys.Scripta T99(2002)34 [hep-ph/0202242]

JB, Acta Phys. Slov. 56(2005)305 [hep-ph/0511076]

η pη

π⁺p_π⁺ π⁻pπ⁻

π⁰ p_π⁰

s = (p_π⁺+ p_π−)²= (pη− pπ⁰)² t = (p_π−+ p_π0)² = (pη− pπ⁺)² u = (p_π⁺+ p_π0)²= (pη − pπ⁻)² s+ t + u = m²_η+ 2m_π²+ + m_π²0 ≡ 3s0.

hπ⁰π⁺π⁻out|ηi = i (2π)⁴δ⁴(pη− pπ⁺− pπ⁻− pπ⁰) A(s, t, u) . hπ⁰π⁰π⁰out|ηi = i (2π)⁴ δ⁴(pη− p1− p2− p3) A(s₁, s₂, s₃) A(s₁, s₂, s₃) = A(s₁, s₂, s₃) + A(s₂, s₃, s₁) + A(s₃, s₁, s₂) Obervables: Γ(η → π⁺π⁻π⁰) and r = ^Γ(η→π⁰^π⁰^π⁰⁾

(6)

Definitions: Dalitz plot

x = √

3T₊− T−

Q_η =

√3

2m_ηQ_η(u − t) y = 3T0

Q_η − 1 = 3 ((mη− m^π^o)²− s)

2mηQ_η − 1^iso= 3

2mηQ_η (s₀− s) Qη = mη − 2mπ⁺− mπ⁰

Tⁱ is the kinetic energy of pion πⁱ z = 2

3 X

i=1,3

3Ei − m^η m_η− 3mπ⁰

2

Ei is the energy of pion πⁱ

|M|² = A²₀ 1 + ay + by²+ dx²+ fy³+ gx²y+ · · ·

|M|² = A²₀(1 + 2αz + · · · )

Note: neutral, next order: x and y appear separately

(7)

Relations

Expand amplitudes and use isospin: JB, Ghorbani, arXiv:0709.0230

M(s, t, u) = A

1 + ˜a(s − s⁰) + ˜b(s − s⁰)²+ ˜d(u − t)²+ · · · M(s, t, u) = A

3 + (˜b+ 3˜d)

(s − s0)²+ (t − s0)²+ (u − s0)²

Gives relations (Rη = (2mηQ_η)/3) a = −2RηRe(˜a) , b= R_η²

|˜a|²+ 2Re(˜b)

, d = 6R_η²Re(˜d) . α = 1

2R_η²Re˜b + 3˜d = 1

4 d + b − Rη²|˜a|² ≤ 1 4

d+ b −1 4a²

equality if Im(˜a) = 0

(8)

Relations

Consequences:

Relations between the charged and neutral decay Relations between r and Dalitz plot

(see alsoGasser, Leutwyler, Nucl. Phys. B 250 (1985) 539) If you can calculate Im(˜a) then relation:

nonrelativistic pion EFT

Schneider, Kubis and Ditsche, JHEP 1102 (2011) 028 [1010.3946].

(9)

Definitions: Dalitz plot

0.08 0.1 0.12 0.14 0.16

t [GeV2]

mpiav mpidiff u=t s=u t-threshold u theshold s threshold x=y=0

x variation:

vertical y variation:

parallel to t = u

(10)

Experiment: Decay rates

Width: determined from Γ(η → γγ) and Branching ratios Using the PDG12 partial update 2013 numbers

Γ(η → π⁺π⁻π⁰) = 300 ± 12 eV (inJB,Ghorbani 295 ± 17 eV)

r: 1.426 ± 0.026 (our fit) 1.48 ± 0.05 (our average)

(11)

Experiment: charged

Exp. a b d f

WASA (prel) −1.104(3) 0.144(3) 0.073(3) 0.153(6) KLOE (prel) −1.074(23)(3) 0.179(27)(8) 0.059(25)(10) 0.089(58)

KLOE −1.090(5)(⁺⁸−19) 0.124(6)(10) 0.057(6)(

+7

−16) 0.14(1)(2) Crystal Barrel −1.22(7) 0.22(11) 0.06(4) (input)

Layter et al. −1.08(14) 0.034(27) 0.046(31) Gormley et al. −1.17(2)(21) 0.21(3) 0.06(4)

Crystal Barrel: d input, but a and b insensitive to d

Large correlations: KLOE:

a b d f

a 1 −0.226 −0.405 −0.795

b 1 0.358 0.261

d 1 0.113

f 1

(12)

Experiment: charged

But very good agreement:

-1 -0.8 -0.6

-0.4 -0.2 0

0.2 0.4

0.6 0.8 1 -1 -0.5

0 0.5

1 0

0.5 1 1.5 2 2.5 3

KLOE 08 KLOE prel WASA prel

y

x

(13)

Experiment: neutral

See talks by Kupsc and Unverzagt

Exp. α

GAMS2000 −0.022 ± 0.023

SND −0.010 ± 0.021 ± 0.010 Crystal Barrel −0.052 ± 0.017 ± 0.010 Crystal Ball (BNL) −0.031 ± 0.004

WASA/CELSIUS −0.026 ± 0.010 ± 0.010 KLOE −0.0301 ± 0.0035^+0.0022_−0.0035 WASA@COSY −0.027 ± 0.008 ± 0.005 Crystal Ball (MAMI-B) −0.032 ± 0.002 ± 0.002 Crystal Ball (MAMI-C) −0.032 ± 0.003 All experiments in good agreement

(14)

Why is η → 3π interesting?

Pions are in I = 1 state =⇒ A∼ (m^u− m^d) or αem

αem effect is small

but is there via (mπ⁺− mπ⁰) in kinematics Lowest order vanishes (current algebra) α ˆmand αms small

Baur, Kambor, Wyler, Nucl. Phys. B 460 (1996) 127

η → π⁺π⁻π⁰γ needs to be included directly

Ditsche, Kubis, Meissner, Eur. Phys. J. C 60 (2009) 83 [0812.0344]

Estimates the corrections of α(mu− md) as well Conclusion: at the precision I will discuss not relevant Exception: Cusps and Coulomb at π⁺π⁻ thresholds So η → 3π gives a handle on mu− md

(15)

Chiral Perturbation Theory

Degrees of freedom: Goldstone Bosons from Chiral

Symmetry Spontaneous Breakdown (without η^′) Power counting: Dimensional counting in momenta/masses Expected breakdown scale: Resonances, so Mρ or higher

depending on the channel

Power counting in momenta: Meson loops

p²

1/p²

R d⁴p p⁴

(p²)²(1/p²)²p⁴ = p⁴

(p²) (1/p²) p⁴ = p⁴

(16)

Lagrangians

U(φ) = exp(i√

2Φ/F₀)parametrizes Goldstone Bosons

Φ(x) =





 π⁰

√2 + η8

√6 π⁺ K⁺

π⁻ −π⁰

√2 + η8

√6 K⁰

K⁻ K¯⁰ −2 η8

√6





 .

LO Lagrangian: L2 = ^F₄⁰²{hD^µU^†D^µUi + hχ^†U+ χU^†i} , D_µU = ∂_µU− irµU + iUl_µ,

left and right external currents: r (l )µ= vµ+ (−)a^µ

Scalar and pseudoscalar external densities: χ = 2B₀(s + ip) quark masses via scalar density: s = M + · · ·

hAi = TrF(A)

(17)

Lagrangians

L4= L₁hDµU^†D^µUi²+ L₂hDµU^†D_νUihD^µU^†D^νUi +L₃hD^µU^†D_µUD^νU^†D_νUi + L4hD^µU^†D_µUihχ^†U + χU^†i +L₅hD^µU^†DµU(χ^†U + U^†χ)i + L6hχ^†U+ χU^†i²

+L₇hχ^†U− χU^†i²+ L₈hχ^†Uχ^†U + χU^†χU^†i

−iL9hFµν^RD^µUD^νU^†+ F_µν^L D^µU^†D^νUi

+L₁₀hU^†F_µν^RUF^Lµνi + H1hFµν^R F^Rµν+ F_µν^L F^Lµνi + H2hχ^†χi L_i: Low-energy-constants (LECs)

Hi: Values depend on definition of currents/densities These absorb the divergences of loop diagrams: Li → L^ri

Renormalization: order by order in the powercounting

(18)

Lagrangians

Lagrangian Structure:

2 flavour 3 flavour 3+3 PQChPT p² F, B 2 F₀, B₀ 2 F₀, B₀ 2 p⁴ l_i^r, h^r_i 7+3 L^r_i, H_i^r 10+2 ˆL^r_i, ˆH_i^r 11+2 p⁶ c_i^r 52+4 C_i^r 90+4 K_i^r 112+3 p²: Weinberg 1966

p⁴: Gasser, Leutwyler 84,85

p⁶: JB, Colangelo, Ecker 99,00

(19)

Chiral Logarithms

The main predictions of ChPT:

Relates processes with different numbers of pseudoscalars Chiral logarithms (and perturbative FSI,. . . )

m_π² = 2B ˆm+ 2B ˆm F

2 1

32π²log(2B ˆm)

µ² + 2l₃^r(µ)

+ · · ·

M² = 2B ˆm

B 6= B⁰, F 6= F⁰ (two versus three-flavour)

(20)

LECs and µ

l₃^r(µ)

¯li = 32π²

γ_i l_i^r(µ) − logM_π² µ² . Independent of the scale µ.

For 3 and more flavours, some of the γi = 0: L^r_i(µ) µ :

m_π, mK: chiral logs vanish pick larger scale

1 GeV then L^r₅(µ) ≈ 0 large Nc arguments????

compromise: µ = m_ρ= 0.77 GeV

(21)

Expand in what quantities?

Expansion is in momenta and masses But is not unique: relations between masses (Gell-Mann–Okubo) exists

Express orders in terms of physical masses and quantities (Fπ, FK)?

Express orders in terms of lowest order masses?

E.g. s + t + u = 2m²_π+ 2m²_K in πK scattering

Relative sizes of order p², p², p⁴,. . . can vary considerably I prefer physical masses

Thresholds correct

Chiral logs are from physical particles propagating

(22)

Expand in what quantities?

Expansion is in momenta and masses But is not unique: relations between masses (Gell-Mann–Okubo) exists

Express orders in terms of physical masses and quantities (Fπ, FK)?

Express orders in terms of lowest order masses?

E.g. s + t + u = 2m²_π+ 2m²_K in πK scattering

Relative sizes of order p², p², p⁴,. . . can vary considerably I prefer physical masses

Thresholds correct

Chiral logs are from physical particles propagating

(23)

LECs

Some combinations of order p⁶ LECs are known as well:

curvature of the scalar and vector formfactor, two more combinations from ππ scattering (implicit in b₅ and b₆) General observation:

Obtainable from kinematical dependences: known Only via quark-mass dependence: poorely known

(24)

C

_i^r

Most analysis use (i.e. almost all of mine):

C_i^r from (single) resonance approximation

π π

ρ, S

→ q² π π

|q²| << m²_ρ, m²_S

=⇒

C_i^r

Motivated by large Nc: large effort goes in this

Ananthanarayan, JB, Cirigliano, Donoghue, Ecker, Gamiz, Golterman, Kaiser, Knecht, Peris, Pich, Prades, Portoles, de Rafael,. . .

(25)

C

_i^r

LV = −1

4hVµνV^µνi +1

2m²_VhVµV^µi − fV

2√

2hVµνf₊^µνi

−igV

2√

2hVµν[u^µ, u^ν]i + fχhVµ[u^µ, χ−]i LA = −1

4hAµνA^µνi +1

2m²_AhAµA^µi − fA

2√

2hAµνf₋^µνi LS = 1

2h∇^µS∇µS− MS²S²i + cdhSu^µu_µi + cmhSχ⁺i Lη^′ = 1

2∂µP1∂^µP1−1

2M_η²′P₁²+ i ˜dmP1hχ−i .

fV = 0.20, fχ= −0.025, gV = 0.09, cm= 42 MeV, cd= 32 MeV,

˜dm= 20 MeV, mV = mρ= 0.77 GeV, mA = ma1= 1.23 GeV, m_S = 0.98 GeV, mP1 = 0.958 GeV

fV, gV, fχ, fA: experiment

(26)

C

_i^r

Problems:

Weakest point in the numerics

However not all results presented depend on this Unknown so far: C_i^r in the masses/decay constants and how these effects correlate into the rest

No µ dependence: obviously only estimate What we do/did about it:

Vary resonance estimate by factor of two

Vary the scale µ at which it applies: 600-900 MeV Check the estimates for the measured ones

Again: kinematic can be had, quark-mass dependence difficult

(27)

L

^r_i

and C

_i^r

Full NNLO fits of the L^r_i

Amor´os,JB,Talavera, 2000, 2001(fit 10) simple C_i^r

JB, Jemos, 2011

simple C_i^r

JB,Ecker,2014, to be published

Continuum fit with more input for C_i^r

Numerics presented for η → 3π is with fit 10

JB,Ghorbani, 2007

Would expect no major changes from that

(28)

Older reviews Model independent ChPT η→ 3π in ChPT

LO LO and NLO NNLO η→ π⁰γγ Conclusions

Lowest order

ChPT:Cronin 67: A(s, t, u) = B₀(mu− m^d) 3√

3F_π²

1 + 3(s − s0) m_η²− m²π

with Q² ≡ _m^m2²^s^{− ˆ}^m²

d−m²u or R≡ _m^m_d^s_−m^{− ˆ}^m_u mˆ = ¹₂(m_u+ m_d) A(s, t, u) = 1

Q² m²_K

m²_π(m²_π− mK²)M(s, t, u) 3√

3F_π² , A(s, t, u) =

√3

4R M(s, t, u)

LO: M(s, t, u) = 3s − 4mπ²

m²_η− m²π

M(s, t, u) = 1 F_π²

4 3m²_π− s

(29)

Lowest order

ChPT:Cronin 67: A(s, t, u) = B₀(mu− m^d) 3√

3F_π²

1 + 3(s − s0) m_η²− m²π

with Q² ≡ _m^m2²^s^{− ˆ}^m²

d−m²u or R≡ _m^m_d^s_−m^{− ˆ}^m_u mˆ = ¹₂(m_u+ m_d) A(s, t, u) = 1

Q² m²_K

m²_π(m²_π− mK²)M(s, t, u) 3√

3F_π² ,

A(s, t, u) =

√3

4R M(s, t, u)

LO: M(s, t, u) = 3s − 4mπ²

m²_η− m²π

M(s, t, u) = 1 F_π²

4 3m²_π− s

(30)

η → 3π: p

²

and p

⁴

Γ (η → 3π) ∝ |A|² ∝ Q⁻⁴ allows a PRECISE measurement Q² form lowest order mass relation: Q ≈ 24

=⇒ Γ(η → π⁺π⁻π⁰)LO≈ 66 eV

m²_K+ − m_K²0 ∼ Q⁻² at NNLO: Q= 20.0 ± 1.5

=⇒ Γ(η → π⁺π⁻π⁰)LO≈ 140 eV

At order p⁴ Gasser-Leutwyler 1985: Z

dLIPS|A2+ A₄|² Z

dLIPS|A²|²

= 2.4 ,

(LIPS=Lorentz invariant phase-space)

Major source: large S-wave final state rescattering Experiment: 300 ± 12 eV (PDG 2012/13)

(31)

η → 3π: p

²

and p

⁴

=⇒ Γ(η → π⁺π⁻π⁰)LO≈ 66 eV

m²_K+ − m_K²0 ∼ Q⁻² at NNLO: Q= 20.0 ± 1.5

=⇒ Γ(η → π⁺π⁻π⁰)LO≈ 140 eV

dLIPS|A2+ A₄|² Z

dLIPS|A²|²

= 2.4 ,

(32)

η → 3π: p

²

and p

⁴

=⇒ Γ(η → π⁺π⁻π⁰)LO≈ 66 eV

m²_K+ − m_K²0 ∼ Q⁻² at NNLO: Q= 20.0 ± 1.5

=⇒ Γ(η → π⁺π⁻π⁰)LO≈ 140 eV

dLIPS|A2+ A₄|² Z

dLIPS|A²|²

= 2.4 ,

(33)

η → 3π: LO, NLO, NNLO, NNNLO,. . .

IN Gasser,Leutwyler, 1985(√

2.4 = 1.55):

about half: ππ-rescattering other half: everything else

ππ-rescattering important Roiesnel, Truong, 1981

Dispersive approach (next talk): resum all ππ assume rescattering + rest separable:

LO NLO NNLO

· · ·

NLO NNLO

· · ·

NNLO

· · ·

→ ππ-rescattering

dispersive does this all the way

↑ Other effects

(34)

Why look at it this way?

LO NLO NNLO

· · ·

NLO NNLO

· · ·

NNLO

· · ·

↑ Other effects

δπ = 0.3, δO = 0.3 LO = 1

NLO = δπ+ δO = 0.6

NNLO = δ²_π+ δπδO + δ²_O = 0.27 Squared: 1 → 2.6 → 3.5

Underlying other is: 1 + 0.3 + 0.09

Goal: remove dispersive from ChPT, then add again via dispersion relations (but now all boxes)

Problem: Separation is not trivial

(35)

Why look at it this way?

LO NLO NNLO

· · ·

NLO NNLO

· · ·

NNLO

· · ·

↑ Other effects

δπ = 0.3, δO = 0.3 LO = 1

NLO = δπ+ δO = 0.6

(36)

Why look at it this way?

LO NLO NNLO

· · ·

NLO NNLO

· · ·

NNLO

· · ·

↑ Other effects

δπ = 0.3, δO = 0.3 LO = 1

NLO = δπ+ δO = 0.6

(37)

Two Loop Calculation: why?

In K_ℓ4 dispersive gave about half of p⁶ in amplitude Same order in ChPT as masses for consistency check on m_u/md

Check size of 3 pion dispersive part

At order p⁴ unitarity about half of correction Technology exists:

Two-loops: Amor´os,JB,Dhonte,Talavera,. . .

Dealing with the mixing π⁰-η: Amor´os,JB,Talavera 01

Done: JB, Ghorbani, arXiv:0709.0230

Dealing with the mixing π⁰-η: extended to η → 3π

(38)

Two Loop Calculation: why?

In K_ℓ4 dispersive gave about half of p⁶ in amplitude Same order in ChPT as masses for consistency check on m_u/md

Check size of 3 pion dispersive part

At order p⁴ unitarity about half of correction Technology exists:

Two-loops: Amor´os,JB,Dhonte,Talavera,. . .

Dealing with the mixing π⁰-η: Amor´os,JB,Talavera 01

Done: JB, Ghorbani, arXiv:0709.0230

Dealing with the mixing π⁰-η: extended to η → 3π

(39)

Diagrams

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Include mixing, renormalize, pull out factor ^√_4R³, . . . Two independent calculations (comparison lots of work) You have to carefully define which LO (Mor M) You have to carefully define which NLO

Integrals only in numerical form: (g) is the hardest one

(40)

η → 3π: M(s, t = u)

-10 0 10 20 30 40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−M(s,t,u=t)

s [GeV²] Li fit 10 and Ci

Re p² Re p²+p⁴ Re p²+p⁴+p⁶ Im p⁴ Im p⁴+p⁶

Along t = u

-2 0 2 4 6 8

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−M(s,t,u=t)

Re p⁶ pure loops Re p⁶ Li r Re p⁶ Ci r sum p⁶

Along t = u parts

(41)

η → 3π: M(s, t = u)

-10 0 10 20 30 40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

M(s,t,u=t)

s [GeV²] Li = Ci = 0

Re p² Re p²+p⁴ Re p²+p⁴+p⁶ Im p⁴ Im p⁶

Along t = u L^r_i = C_i^r = 0

-10 0 10 20 30 40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−M(s,t,u=t)

Re p²+p⁴ µ= 0.6 GeV µ= 0.9 GeV Re p²+p⁴+p⁶ µ= 0.6 GeV µ= 0.9 GeV

Along t = u: µ dependence I.e. where C_i^r(µ) estimated

(42)

Neutral decay

A²₀ α

LO 1090 0.000

NLO 2810 0.013

NLO (L^r_i = 0) 2100 0.016

NNLO 4790 0.013

NNLOq 4790 0.014

NNLO (C_i^r = 0) 4140 0.011 NNLO (L^r_i = C_i^r = 0) 2220 0.016

dispersive (KWW) — −(0.007—0.014) tree dispersive — −0.0065 absolute dispersive — −0.007

Borasoy — −0.031

error 160 0.032

experiment: α = −0.032 with small error NNLO ChPT gets a₀⁰ in ππ correct